Harmonic analysis for rank-1 Randomised Horn Problems

Abstract

The randomised Horn problem, in both its additive and multiplicative version, has recently drawn increasing interest. Especially, closed analytical results have been found for the rank-1 perturbation of sums of Hermitian matrices and products of unitary matrices. We will generalise these results to rank-1 perturbations for products of positive definite Hermitian matrices and prove the other results in a new unified way. Our ideas work along harmonic analysis for matrix groups via spherical transforms that have been successfully applied in products of random matrices in the past years. In order to achieve the unified derivation of all three cases, we define the spherical transform on the unitary group and prove its invertibility.

arXiv comment24 pages

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Cited by in corpus (8)

• Stability of large complex systems with heterogeneous relaxation dynamics
• Asymptotic behavior of the multiplicative counterpart of the Harish-Chandra integral and the $S$-transform
• Derivative principles for invariant ensembles
• Dip-ramp-plateau for Dyson Brownian motion from the identity on $U(N)$
• Projections of Orbital Measures and Quantum Marginal Problems
• Cyclic Pólya Ensembles on the Unitary Matrices and their Spectral Statistics
• Product Matrix Processes with Symplectic and Orthogonal Invariance via Symmetric Functions
• One-matrix differential reformulation of two-matrix models